A156690 - OEIS

%I #12 Feb 26 2021 05:48:21

%S 1,1,1,1,-3,1,1,15,15,1,1,-105,525,-105,1,1,945,33075,33075,945,1,1,

%T -10395,3274425,-22920975,3274425,-10395,1,1,135135,468242775,

%U 29499294825,29499294825,468242775,135135,1,1,-2027025,91307341125,-63275987399625,569483886596625,-63275987399625,91307341125,-2027025,1

%N Triangle T(n, k, m) = t(n,m)/( t(k,m) * t(n-k,m) ) with T(n, 0, m) = T(n, n, m) = 1, where t(n, m) = Product_{j=1..n} Product_{i=1..j-1} ( 1 - (m+1)*(i+1) ) and m = 1, read by rows.

%C Row sums are: {1, 2, -1, 32, 317, 68042, -16392913, 59935345472, 443114522425577, 41952026212764267602, -11773681484663891313796273, ...}.

%H G. C. Greubel, <a href="/A156690/b156690.txt">Rows n = 0..30 of the triangle, flattened</a>

%F T(n, k, m) = t(n,m)/( t(k,m) * t(n-k,m) ) with T(n, 0, m) = T(n, n, m) = 1, where t(n, m) = Product_{j=1..n} Product_{i=1..j-1} ( 1 - (m+1)*(i+1) ) and m = 1.

%F T(n, k, m, p, q) = (-p*(m+1))^(k*(n-k)) * (f(n,m,p,q)/(f(k,m,p,q)*f(n-k,m,p,q))) where Product_{j=1..n} Pochhammer( (q*(m+1) -1)/(p*(m+1)), j) for (m, p, q) = (1, 1, 1). - _G. C. Greubel_, Feb 25 2021

%e Triangle begins as:

%e   1;

%e   1,      1;

%e   1,     -3,         1;

%e   1,     15,        15,           1;

%e   1,   -105,       525,        -105,           1;

%e   1,    945,     33075,       33075,         945,         1;

%e   1, -10395,   3274425,   -22920975,     3274425,    -10395,      1;

%e   1, 135135, 468242775, 29499294825, 29499294825, 468242775, 135135, 1;

%t (* First program *)

%t t[n_, k_]:= If[k==0, n!, Product[1 -(i+1)*(k+1), {j,n}, {i,0,j-1}] ];

%t T[n_, k_, m_]:= If[n==0, 1, t[n, m]/(t[k, m]*t[n-k, m])];

%t Table[T[n,k,1], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Feb 25 2021 *)

%t (* Second program *)

%t f[n_, m_, p_, q_]:= Product[Pochhammer[(q*(m+1) -1)/(p*(m+1)), j], {j,n}];

%t T[n_, k_, m_, p_, q_]:= (-p*(m+1))^(k*(n-k))*(f[n,m,p,q]/(f[k,m,p,q]*f[n-k,m,p,q]));

%t Table[T[n,k,1,1,1], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Feb 25 2021 *)

%o (Sage)

%o @CachedFunction

%o def f(n, m, p, q): return product( rising_factorial( (q*(m+1)-1)/(p*(m+1)), j) for j in (1..n))

%o def T(n,k,m,p,q): return (-p*(m+1))^(k*(n-k))*(f(n,m,p,q)/(f(k,m,p,q)*f(n-k,m,p,q) ))

%o flatten([[T(n,k,1,1,1) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 25 2021

%o (Magma)

%o f:= func< n,m,p,q | n eq 0 select 1 else m eq 0 select Factorial(n) else (&*[ 1 -(p*i+q)*(m+1): i in [0..j], j in [0..n-1]]) >;

%o T:= func< n,k,m,p,q | f(n,m,p,q)/(f(k,m,p,q)*f(n-k,m,p,q)) >;

%o [T(n,k,1,1,1): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Feb 25 2021

%Y Cf. A007318 (m=0), this sequence (m=1), A156691 (m=2), A156692 (m=3).

%Y Cf. A156693, A156696, A156722.

%K sign,tabl

%O 0,5

%A _Roger L. Bagula_, Feb 13 2009

%E Edited by _G. C. Greubel_, Feb 25 2021